1. ©K.Cuthbertson and D.Nitzsche 1 FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001) K. Cuthbertson and D. Nitzsche LECTURE Dynamic Hedging and the Greeks 1/9/2001

2. ©K.Cuthbertson and D.Nitzsche Topics Dynamic (Delta) Hedging The Greeks BOPM and the Greeks

3. ©K.Cuthbertson and D.Nitzsche 3 Dynamic Hedging

4. ©K.Cuthbertson and D.Nitzsche Dynamic (Delta) Hedging Suppose we have written a call option for C 0 =10.45 (with K=100,  = 20%, r=5%, T=1) when the current stock price is S 0 =100 and  0 = 0.6368 At t=0, to hedge the call we buy  0 = 0.6368 shares at So = 100 at a cost of $63.68. hence we need to borrow (i.e. go into debt) Debt, D 0 =  0 S 0 - C 0 = 63.68 – 10.45 = $53.23

5. ©K.Cuthbertson and D.Nitzsche Dynamic Delta Hedging At t = 1 the stock price has fallen to S 1 = 99 with  1 = 0.617. You therefore sell (  1 -  0 ) shares at S 1 generating a cash inflow of $1.958 which can be used to reduce your debt so that your debt position at t=1 is 53.23 - 1.958 = 51.30 The value of your hedge portfolio at t = 1 (including the market value of your written call): V 1 = = Value of shares held - Debt - Call premium = = 0.0274 (approx zero) But as S falls (say) then you sell on a falling marker ending up with positive debt

6. ©K.Cuthbertson and D.Nitzsche Dynamic Delta Hedging OPTION ENDS UP OUT-OF-THE-MONEY (  T = 0 shares) $ Net cost at T: D T = 10.19 % Net cost at T: ( D T - C 0 ) / C 0 = 2.46% OPTION ENDS UP IN THE-MONEY (  T = 1 share) $ Net cost at T: D T – K = 111.29 – 100 = 11.29 % Net cost at T: ( D T – K - C 0 ) / C 0 = 8.1% % Cost of the delta hedge = risk free rate %Hedge Performancer = sd( D T e -rT - C 0 ) / C 0

7. ©K.Cuthbertson and D.Nitzsche 7 THE GREEKS

8. ©K.Cuthbertson and D.Nitzsche 8 Figure 9.2 : Delta and gamma : long call Delta Gamma Stock Price (K = 50)

9. ©K.Cuthbertson and D.Nitzsche 9 THE GREEKS: A RISK FREE HOLIDAY ON THE ISLANDS Gamma and Lamda df  .dS +(1/2)  (dS) 2 +  dt +  dr +  d 

10. ©K.Cuthbertson and D.Nitzsche 10 HEDGING WITH THE GREEKS: Gamma Neutral Portfolio  = gamma of existing portfolio  T = gamma of “new” options  port = N T  T +  = 0 therefore “buy” : N T = -  /  T “new” options Vega Neutral Portfolio Similarly : N  = -  /  T “new” options

11. ©K.Cuthbertson and D.Nitzsche 11 HEDGING WITH THE GREEKS ORDER OF CALCULATIONS 1) Make existing portfolio either vega or gamma neutral (or both simultaneously, if required in the hedge) by buying/selling “other” options. Call this portfolio-X 2) Portfolio-X is not delta neutral. Now make portfolio-X delta neutral by trading only the underlying stocks (can’t trade options because this would “break the gamma/vega neutrality).

12. ©K.Cuthbertson and D.Nitzsche 12 Hedging With The Greeks: A Simple Example Portfolio–A: is delta neutral but  = -300. A Call option “Z” with the same underlying (e.g. stock) has a delta = 0.62 and gamma of 1.5. How can you use Z to make the overall portfolio gamma and delta neutral? We require: n z  z +  = O n z = -  /  z = -(-300)/1.5 = 200 implies 200 long contracts in Z The delta of this ‘new’ portfolio is now  = n z.  z = 200(0.62) = 124 Hence to maintain delta neutrality you must short 124 units of the underlying.

13. ©K.Cuthbertson and D.Nitzsche 13 BOPM and the Greeks

14. ©K.Cuthbertson and D.Nitzsche 14 Figure 9.5 : BOPM lattice Index, j Time, t 1,02,03,04,0 0,0 1,1 2,2 3,3 4,4 2,13,14,1 3,24,2 4,3 1 0234

15. ©K.Cuthbertson and D.Nitzsche 15 BOPM and the Greeks Gamma S* = (S 22 + S 21 )/2 and in the lower part, S** = (S 21 + S 20 )/2. Hence their difference is: [9.32] = ] /2 =

16. ©K.Cuthbertson and D.Nitzsche 16 End of Slides